Volume 6 (2000), No. 1 (January), pp. 1--157

I.I. Makarov. Dynamical Entropy for Markov Operators. J. Dynam. Control Systems 6 (2000), no. 1, 1--11.

Let (X,\Sigma,\mu) be a probability space. An operator P : L^{1}(X,\Sigma,\mu) \to L^{1}(X,\Sigma,\mu) is called a Markov operator if it satisfies the following conditions:

(1) P{\bold 1}={\bold 1};

(2) \forall f \in L^{1}(X,\Sigma,\mu), f\ge 0 \Rightarrow Pf \ge 0;

(3) \int\limits_{X} Pf\,d\mu=\int\limits_{X} f\,d\mu.

The purpose of this paper is to extend the notion of entropy to the class of Markov operators. The relationship between the Kolmogorov--Sinai entropy and new defined one will be examined.

E.V. Zhuzhoma, V.S. Medvedev. Mappings with Three Homeomorphism Intervals that Change the Orientation of One of Them. J. Dynam. Control Systems 6 (2000), no. 1, 13--19.

In this paper, we consider one-to-one piecewise-homeomorphic mappings of a circle that have three homeomorphism intervals and change the orientation of one of the intervals. Moreover, it is assumed that the mappings themselves as well as their extensions to endpoints have no periodic orbits (such mappings arise in a natural way in the study of flows without periodic trajectories and separatrix connections on nonorientable surfaces). We obtain a classification (from the conjugation viewpoint) of such mappings. A key result consists in the proof of existence of a wandering interval. Also, we prove that any mapping under consideration has exactly one invariant normalized measure.

V.A. Kaimanovich. Ergodic Properties of the Horocycle Flow and Classification of Fuchsian Groups. J. Dynam. Control Systems 6 (2000), no. 1, 21--56.

The paper is devoted to the study of the basic ergodic properties (ergodicity and conservativity) of the horocycle flow on surfaces of constant negative curvature with respect to the Liouville invariant measure. We give several criteria for ergodicity and conservativity and connect them with the classification of the associated Fuchsian groups. Special attention is given to covering surfaces. In particular, we show that normal subgroups of divergent-type Fuchsian groups provide natural examples for the strictness of a number of inclusions in the classification of Fuchsian groups. This paper in a sense complements and continues earlier work by A.N. Starkov ``Fuchsian groups from the dynamical viewpoint'' published in the same journal in 1995.

A.A. Bolibruch. On Orders of Movable Poles of the Schlesinger Equation. J. Dynam. Control Systems 6 (2000), no. 1, 57--73.

We treat here movable singular points of the Schlesinger equation of isomonodromic deformations as the points of jumping of the splitting type of the vector bundle constructed from the initial Fuchsian equation. We show that solutions to the Schlesinger equation have at these points poles of order two and we also illustrate the results mentioned above by means of some Fuchsian system of two differential equations with four singular points.

G.S. Chakvetadze. Stochastic Stability in a Model of Drilling. J. Dynam. Control Systems 6 (2000), no. 1, 75--95.

A random perturbation of a dynamical system serving as a model for the drilling process is studied. For this perturbation the existence of absolutely continuous stationary distributions is proved. It is shown that these distributions tend to a unique absolutely continuous invariant probability of the nonperturbed system as the magnitude of perturbation tends to zero.

V.Z. Grines. On Topological Classification of A-Diffeomorphisms of Surfaces. J. Dynam. Control Systems 6 (2000), no. 1, 97--126.

The paper contains a survey of works of the author devoted to topological classification of A-diffeomorphisms on surfaces. In the survey, the following problems are considered.

(1) Topological classification of one-dimensional attractors and repellers by means of automorphisms of fundamental group of supports and its representation by hyperbolic homeomorphisms is given.

(2) Asymptotic behavior of preimages of stable and unstable manifolds of points belonging to exteriorly situated one-dimensional basic sets on universal covering is investigated (it is proved that if A-diffeomorphism is structurally stable, then these preimages boundedly deviate from geodesic with the same asymptotic direction).

(3) Topological classification of an important class of structurally stable diffeomorphisms is obtained.

S.Kh. Aranson. Qualitative Properties of Foliations on Closed Surfaces. J. Dynam. Control Systems 6 (2000), no. 1, 127--157.

The paper contains a survey of the author's results obtained at last ten years on a research of foliations with singularities on closed surfaces. The following problems of the qualitative theory of foliations are considered.

(1) Generalization of the Poincare--Bendixon theory.

(2) Kneser problem and estimation of the number of quasiminimal sets.

(3) Anosov problem about interrelation between geodesics and asymptotic behavior of leaves of foliations.

(4) Topological classification of supertransitive foliations.